Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 24 doc

For example, if a one-year bond promises you a return of 10 percent and the expected inflation rate is 4 percent, the expected realreturn on your bond is , or 5.8 percent.. If the annual

V A L U I N G D E B T

flows and discount them at the opportunity cost of capital Therefore, if a bond produces cash flows of

C dollars per year for N years and is then repaid at its face value ($1,000), the present value is

where are the appropriate discount rates for the cash flows to be received by the bondowners in periods

That is correct as far as it goes but it does not tell us anything about what determines the discount

rates For example,

• In 1945 U.S Treasury bills offered a return of 4 percent: At their 1981 peak they offered a turn of over 17 percent Why does the same security offer radically different yields at differenttimes?

re-• In mid-2001 the U.S Treasury could borrow for one year at an interest rate of 3.4 percent, but ithad to pay nearly 6 percent for a 30-year loan Why do bonds maturing at different dates offer dif-

ferent rates of interest? In other words, why is there a term structure of interest rates?

• In mid-2001 the United States government could issue long-term bonds at a rate of nearly

6 percent But even the most blue-chip corporate issuers had to pay at least 50 basis points (.5 percent) more on their long-term borrowing What explains the premium that firms have

We do not believe that ignorance is desirable even when it is harmless At least you ought to be

able to read the bond tables in The Wall Street Journal and talk to investment bankers about the

prices of recently issued bonds More important, you will encounter many problems of bond pricingwhere there are no similar instruments already traded How do you evaluate a private placement with

a custom-tailored repayment schedule? How about financial leases? In Chapter 26 we will see thatthey are essentially debt contracts, but often extremely complicated ones, for which traded bondsare not close substitutes Many companies, notably banks and insurance firms, are exposed to therisk of interest rate fluctuations To control their exposure, these companies need to understand howinterest rates change.1You will find that the terms, concepts, and facts presented in this chapter areessential to the analysis of these and other practical problems

We start the chapter with our first question: Why does the general level of interest rates changeover time? Next we turn to the relationship between short- and long-term interest rates We considerthree issues:

• Each period’s cash flow on a bond potentially needs to be discounted at a different interest rate,but bond investors often calculate the yield to maturity as a summary measure of the interest rate

on the bond We first explain how these measures are related

Indexed Bonds and the Real Rate of Interest

In Chapter 3 we drew the distinction between the real and nominal rate of interest

Most bonds promise a fixed nominal rate of interest The real interest rate that you

receive depends on the inflation rate For example, if a one-year bond promises you

a return of 10 percent and the expected inflation rate is 4 percent, the expected realreturn on your bond is , or 5.8 percent Since future inflationrates are uncertain, the real return on a bond is also uncertain For example, if in-flation turns out to be higher than the expected 4 percent, the real return will be

lower than 5.8 percent.

You can nail down a real return; you do so by buying an indexed bond whose

payments are linked to inflation Indexed bonds have been around in many tries for decades, but they were almost unknown in the United States until 1997when the U.S Treasury began to issue inflation-indexed bonds known as TIPs(Treasury Inflation-Protected Securities).2

coun-The real cash flows on TIPs are fixed, but the nominal cash flows (interest andprincipal) are increased as the Consumer Price Index increases For example, sup-pose that the U.S Treasury issues 3 percent 20-year TIPs at a price of 100 If duringthe first year the Consumer Price Index rises by (say) 10 percent, then the couponpayment on the bond would be increased by 10 percent to percent.And the final payment of principal would also be increased in the same proportion

price and holds it to maturity can be assured of a real yield of 3 percent

As we write this in the summer of 2001, long-term TIPs offer a yield of 3.46

per-cent This yield is a real yield: It measures how much extra goods your investment

would allow you to buy The 3.46 percent yield on TIPs was about 2.3 percent lessthan on nominal Treasury bonds If the annual inflation rate proves to be higherthan 2.3 percent, you will earn a higher return by holding long-term TIPs; if the in-flation rate is lower than 2.3 percent, the reverse will be true

What determines the real interest rate that investors demand? The classical omist’s answer to this question is summed up in the title of Irving Fisher’s great

econ-book: The Theory of Interest: As Determined by Impatience to Spend Income and

11.1 ⫻ 1002 ⫽ 110

11.1 ⫻ 32 ⫽ 3.31.10/1.04⫺ 1 ⫽ 058

• Second, we show why a change in interest rates has a greater impact on the price of long-termloans than on short-term loans

• Finally, we look at some theories that explain why short- and long-term interest rates differ

To close the chapter we shift the focus to corporate bonds and examine the risk of default and its fect on bond prices

ef-24.1 REAL AND NOMINAL RATES OF INTEREST

2 In 1988 Franklin Savings Association had issued a 20-year bond whose interest (but not principal) was tied to the rate of inflation Since then a trickle of companies has also issued indexed bonds.

3 August M Kelley, New York, 1965; originally published in 1930.

the supply and demand for capital The supply depends on people’s willingness to

save.4The demand depends on the opportunities for productive investment

For example, suppose that investment opportunities generally improve Firms

have more good projects, so they are willing to invest more than previously at any

interest rate Therefore, the rate has to rise to induce individuals to save the

addi-tional amount that firms want to invest.5Conversely, if investment opportunities

deteriorate, there will be a fall in the real interest rate

Fisher’s theory emphasizes that the required real rate of interest depends on real

phenomena A high aggregate willingness to save may be associated with high

ag-gregate wealth (because wealthy people usually save more), an uneven

distribu-tion of wealth (an even distribudistribu-tion would mean fewer rich people, who do most

of the saving), and a high proportion of middle-aged people (the young don’t need

to save and the old don’t want to—“You can’t take it with you”) Correspondingly,

a high propensity to invest may be associated with a high level of industrial

activ-ity or major technological advances

Real interest rates do change but they do so gradually We can see this by

look-ing at the UK, where the government has issued indexed bonds since 1982 The

col-ored line in Figure 24.1 shows that the (real) yield on these bonds has fluctuated

within a relatively narrow range, while the yield on nominal government bonds

has declined dramatically

Inflation and Nominal Interest Rates

Now let us see what Irving Fisher had to say about inflation and interest rates

Sup-pose that consumers are equally happy with 100 apples today or 105 apples in a

year’s time In this case the real or “apple” interest rate is 5 percent Suppose also

4 Some of this saving is done indirectly For example, if you hold 100 shares of GM stock, and GM

re-tains earnings of $1 per share, GM is saving $100 on your behalf.

5 We assume that investors save more as interest rates rise It doesn’t have to be that way; here is an

ex-ample of how a higher interest rate could mean less saving: Suppose that 20 years hence you will need

$50,000 at current prices for your children’s college expenses How much will you have to set aside

to-day to cover this obligation? The answer is the present value of a real expenditure of $50,000 after 20

years, or The higher the real interest rate, the lower the present value

and the less you have to set aside.

50,000/11 ⫹ real interest rate2 20

Dec 83 Dec 84 Dec 85 Dec 86 Dec 87 Dec 88 Dec 89 Dec 90 Dec 91 Dec 92 Dec 93 Dec 94 Dec 95 Dec 96 Dec 97 Dec 98 Dec 99 Dec 00

Real yield on UK indexed bonds

Yield on UK nominal bonds

F I G U R E 2 4 1

The burgundy line shows the real yield

on long-term indexed bonds issued by the UK government The blue line shows the yield on UK government long-term nominal bonds Notice that the real yield has been much more stable than the nominal yield.

that I know the price of apples will increase over the year by 10 percent Then I willpart with $100 today if I am repaid $115 at the end of the year That $115 is needed

to buy me 5 percent more apples than I can get for my $100 today In other words,the nominal, or “money,” rate of interest must equal the required real, or “apple,”rate plus the prospective rate of inflation.6A change of 1 percent in the expected in-flation rate produces a change of 1 percent in the nominal interest rate That isFisher’s theory: A change in the expected inflation rate will cause the same change

in the nominal interest rate; it has no effect on the required real interest rate.7

Nominal interest rates cannot be negative; if they were, everyone would prefer

to hold cash, which pays zero interest.8But what about real rates? For example, is

it possible for the money rate of interest to be 5 percent and the expected rate of flation to be 10 percent, thus giving a negative real interest rate? If this happens,you may be able to make money in the following way: You borrow $100 at an in-terest rate of 5 percent and you use the money to buy apples You store the applesand sell them at the end of the year for $110, which leaves you enough to pay offyour loan plus $5 for yourself

in-Since easy ways to make money are rare, we can conclude that if it doesn’t costanything to store goods, the money rate of interest can’t be less than the expectedrise in prices But many goods are even more expensive to store than apples, andothers can’t be stored at all (you can’t store haircuts, for example) For these goods,the money interest rate can be less than the expected price rise

How Well Does Fisher’s Theory Explain Interest Rates?

Not all economists would agree with Fisher that the real rate of interest is fected by the inflation rate For example, if changes in prices are associated withchanges in the level of industrial activity, then in inflationary conditions I mightwant more or less than 105 apples in a year’s time to compensate me for the loss of

unaf-100 today

We wish we could show you the past behavior of interest rates and expected

in-flation Instead we have done the next best thing and plotted in Figure 24.2 the turn on U.S Treasury bills against actual inflation Notice that between 1926 and

re-1981 the return on Treasury bills was below the inflation rate about as often as it

6

We oversimplify If apples cost $1.00 apiece today and $1.10 next year, you need next year to buy 105 apples The money rate of interest is 15.5 percent, not 15 Remember, the exact for- mula relating real and money rates is

where i is the expected inflation rate Thus

In our example, the money rate should be

When we said the money rate should be 15 percent, we ignored the cross-product term i This is

a common rule of thumb because the cross-product term is usually small But there are countries where

i is large (sometimes 100 percent or more) In such cases it pays to use the full formula.

7

The apple example was taken from R Roll, “Interest Rates on Monetary Assets and Commodity Price

Index Changes,” Journal of Finance 27 (May 1972), pp 251–278.

was above The average real interest rate during this period was a mere 0.1 percent.

Since 1981 the return on bills has been significantly higher than the rate of

infla-tion, so that investors have earned a positive real return on their savings

Fisher’s theory states that changes in anticipated inflation produce

correspon-ding changes in the rate of interest But Figure 24.2 offers little evidence of this in

the 1930s and 1940s During this period, the return on Treasury bills scarcely

changed even though the inflation rate fluctuated sharply Either these changes in

inflation were unanticipated or Fisher’s theory was wrong Since the early 1950s,

there appears to have been a closer relationship between interest rates and

infla-tion in the United States.9Thus, for today’s financial managers Fisher’s theory

pro-vides a useful rule of thumb If the expected inflation rate changes, it is a good bet

that there will be a corresponding change in the interest rate

9 This probably reflects government policy, which before 1951 stabilized nominal interest rates The 1951

“accord” between the Treasury and the Federal Reserve System permitted more flexible nominal

inter-est rates after 1951.

24.2 TERM STRUCTURE AND YIELDS TO MATURITY

We turn now to the relationship between short- and long-term rates of interest

Suppose that we have a simple loan that pays $1 at time 1 The present value of this

loan is

Thus we discount the cash flow at , the rate appropriate for a one-period loan

This rate, which is fixed today, is often called today’s one-period spot rate.

If we have a loan that pays $1 at both time 1 and time 2, present value is

Thus the first period’s cash flow is discounted at today’s one-period spot rate andthe second period’s flow is discounted at today’s two-period spot rate The series

of spot rates , etc., is one way of expressing the term structure of interest rates Yield to Maturity

Rather than discounting each of the payments at a different rate of interest, we couldfind a single rate of discount that would produce the same present value Such a rate

is known as the yield to maturity, though it is in fact no more than our old

acquain-tance, the internal rate of return (IRR), masquerading under another name If we call

the yield to maturity y, we can write the present value of the two-year loan as

All you need to calculate y is the price of a bond, its annual payment, and its

ma-turity You can then rapidly work out the yield with the aid of a preprogrammedcalculator

The yield to maturity is unambiguous and easy to calculate It is also the in-trade of any bond dealer By now, however, you should have learned to treat anyinternal rate of return with suspicion.10The more closely we examine the yield tomaturity, the less informative it is seen to be Here is an example

stock-Example It is 2003 You are contemplating an investment in U.S Treasury bonds

and come across the following quotations for two bonds:11

The phrase “5s of ‘08” refers to a bond maturing in 2008, paying annual interest of

5 percent of the bond’s face value The interest payment is called the coupon payment.

In continental Europe coupons are usually paid annually; in the United States theyare usually paid every six months, so the 5s of ‘08 would pay 2.5 percent of face valueevery six months To simplify the arithmetic, we will pretend throughout this chap-ter that all coupon payments are annual When the bonds mature in 2008, bond-holders receive the bond’s face value in addition to the final interest payment.The price of each bond is quoted as a percent of face value Therefore, if facevalue is $1,000, you would have to pay $852.11 to buy the bond and your yieldwould be 8.78 percent Letting 2003 be , 2004 be , etc., we have the fol-lowing discounted-cash-flow calculation:

Although the two bonds mature at the same date, they presumably were issued at

different times—the 5s when interest rates were low and the 10s when interest rates

were high

Are the 5s of ‘08 a better buy? Is the market making a mistake by pricing these

two issues at different yields? The only way you will know for sure is to calculate

the bonds’ present values by using spot rates of interest: for 2004, for 2005, etc

This is done in Table 24.1

The important assumption in Table 24.1 is that long-term interest rates are

higher than short-term interest rates We have assumed that the one-year interest

rate is , the two-year rate is , and so on When each year’s cash flow

is discounted at the rate appropriate to that year, we see that each bond’s present

value is exactly equal to the quoted price Thus each bond is fairly priced

Why do the 5s have a higher yield? Because for each dollar that you invest in the

5s you receive relatively little cash inflow in the first four years and a relatively

high cash inflow in the final year Therefore, although the two bonds have

identi-cal maturity dates, the 5s provide a greater proportion of their cash flows in 2008

In this sense the 5s are a longer-term investment than the 10s Their higher yield to

maturity just reflects the fact that long-term interest rates are higher than

short-term rates

Notice why the yield to maturity can be misleading When the yield is calculated,

the same rate is used to discount all payments on the bond But in our example

bond-holders actually demanded different rates of return ( , etc.) for cash flows that

oc-curred at different times Since the cash flows on the two bonds were not identical,

the bonds had different yields to maturity Therefore, the yield to maturity on the 5s

of ‘08 offered only a rough guide to the appropriate yield on the 10s of ‘08.12

Measuring the Term Structure

Financial managers who just want a quick, summary measure of interest rates look

in the financial press at the yields to maturity on government bonds Thus managers

will make broad generalizations such as “If we borrow money today, we will have

to pay an interest rate of 8 percent.” But if you wish to understand why different

12

For a good analysis of the relationship between the yield to maturity and spot interest rates, see S M.

Schaefer, “The Problem with Redemption Yields,” Financial Analysts Journal 33 (July–August 1977),

pp 59–67.

bonds sell at different prices, you must dig deeper and look at the separate interestrates for one-year cash flows, for two-year cash flows, and so on In other words, youmust look at the spot rates of interest.

To find the spot interest rate, you need the price of a bond that simply makes one

future payment Fortunately, such bonds do exist They are known as stripped bonds

or strips Strips originated in 1982 when several investment bankers came up with

a novel idea They bought U.S Treasury bonds and reissued their own separatemini-bonds, each of which made only one payment The idea proved to be popu-lar with investors, who welcomed the opportunity to buy the mini-bonds ratherthan the complete package If you’ve got a smart idea, you can be sure that otherswill soon clamber onto your bandwagon It was therefore not long before the Trea-sury issued its own mini-bonds.13The prices of these bonds are shown each day inthe daily press For example, in the summer of 2001, a strip maturing in May 2021cost $316.55 and 20 years later will give the investors a single payment of $1,000

In Figure 24.3 we have used the prices of strips with different maturities to plotthe term structure of spot rates from 1 to 24 years You can see that investors re-quired an interest rate of 3.4 percent from a bond that made a payment only at theend of one year and a rate of 5.8 percent from a bond that paid off only in year 2025

Duration and Bond Volatility

In Chapter 7 we reviewed the historical performance of different security classes

We showed that since 1926 long-term government bonds have provided a higheraverage return than short-term bills, but have also been more variable The stan-

dard deviation of annual returns on a portfolio of long-term bonds was 9.4 percent

compared with a standard deviation of 3.2 percent for bills

Figure 24.4 illustrates why long-term bonds are more variable Each line shows

how the price of a 5-percent bond changes with the level of interest rates You can

see that the price of a longer-term bond is more sensitive to interest rate

fluctua-tions than that of a shorter bond

But what do we mean by long-term and short-term bonds? It is obvious in the

case of strips that make payments in only one year However, a coupon bond that

matures in year 10 makes payments in each of years 1 through 10 Therefore, it is

somewhat misleading to describe the bond as a 10-year bond; the average time to

each cash flow is less than 10 years

Consider the Treasury 6 7/8s of 2006 In mid-2001 these bonds had a present

value of 108.57 percent of face value and yielded 4.9 percent The third and fourth

columns in Table 24.2 show where this present value comes from Notice that the

cash flow in year 5 accounts for only 77.5 percent of the bond’s value The

remain-ing 22.5 percent of the value comes from the earlier cash flows

Interest rate, percent

Proportion of

Year Ct PV(C t) at 4.9% [PV(C t)/V] Total Value ⴛ Time

of the present value of the

6 7/8s of 2006 The final column shows how to calculate

a weighted average of the time

to each cash flow This average

is the bond’s duration.

Bond analysts often use the term duration to describe the average time to each

payment If we call the total value of the bond V, then duration is calculated as

follows:15

For the 6 7/8s of 2006,

The Treasury 4 5/8s of 2006 have the same maturity as the 6 7/8s, but the first fouryears’ coupon payments account for a smaller fraction of the bond’s value In thissense the 4 5/8s are longer bonds than the 6 7/8s The duration of the 4 5/8s is4.574 years

Consider now what happens to the prices of our two bonds as interest rateschange:

1856, National Bureau of Economic Research, New York, 1938.

16For this reason volatility is also called modified duration.

Thus, a 1 percentage-point variation in yield causes the price of the 6 7/8s to change

by 4.22 percent We can say that the 6 7/8s have a volatility of 4.22 percent, while

the 4 5/8s have a volatility of 4.36 percent

Notice that the 4 5/8 percent bonds have the greater volatility and that theyalso have the longer duration In fact, a bond’s volatility is directly related to itsduration:16

In the case of the 6 7/8s,

In Figure 24.4 we showed how bond prices vary with the level of interest rates Eachbond’s volatility is simply the slope of the line relating the bond price to the interestrate You can see this more clearly in Figure 24.5, where the convex curve shows theprice of the 5 percent 30-year bond for different interest rates The bond’s volatility ismeasured by the slope of a tangent to this curve For example, the dotted line in thefigure shows that, if the interest rate is 5 percent, the curve has a slope of 15.4 At thispoint the change in bond price is 15.4 times a change in the interest rate Notice thatthe bond’s volatility changes as the interest rate changes Volatility is higher at lowerinterest rates (the curve is steeper), and it is lower at higher rates (the curve is flatter)

Volatility 1percent2 ⫽4.4241.049⫽ 4.22Volatility 1percent2 ⫽1duration⫹ yield

Managing Interest Rate Risk

Volatility is a useful, summary measure of the likely effect of a change in interest

rates on the value of a bond The longer a bond’s duration, the greater is its

volatil-ity In Chapter 27 we will make use of this relationship between duration and

volatility to describe how firms can protect themselves against interest rate

changes Here is an example that should give you a flavor of things to come

Suppose your firm has promised to make pension payments to retired

employ-ees The discounted value of these pension payments is $1 million; therefore, the

firm puts aside $1 million in the pension fund and invests the money in

govern-ment bonds So the firm has a liability of $1 million and (through the pension fund)

an offsetting asset of $1 million But, as interest rates fluctuate, the value of the

pen-sion liability will change and so will the value of the bonds in the penpen-sion fund

How can the firm ensure that the value of the bonds in the fund is always sufficient

to meet the liabilities? Answer: By making sure that the duration of the bonds is

al-ways the same as the duration of the pension liability

A Cautionary Note

Bond volatility measures the effect on bond prices of a shift in interest rates For

ex-ample, we calculated that the 6 7/8s had a volatility of 4.22 This means a 1

percentage-point change in interest rates leads to a 4.22 percent change in bond price:

This relationship is sometimes called a one-factor model of bond returns; it tells us

how each bond’s price changes in response to one factor—a change in the overall

level of interest rates One-factor models have proved very useful in helping firms

to understand how they are affected by interest-rate changes and how they can

protect themselves against these risks

If the yields on all Treasury bonds moved in precise lockstep, then changes in

the price of each bond would be exactly proportional to the bond’s duration For

example, the price of a long-term bond with a duration of 20 years would always

rise or fall twice as much as the price of a medium-term bond with a duration of 10

years However, Figure 24.6 illustrates that short- and long-term interest rates do

Change in bond price⫽ 4.22 ⫻ change in interest rates

Interest rate, percent

is steeper) and lower at higher rates (the curve is flatter).

not always move in perfect unison Between 1992 and 2000 short-term interest rates

nearly doubled while long-term rates declined As a result, the term structure,which initially sloped steeply upward, shifted to a downward slope Becauseshort- and long-term yields do not move in parallel, one-factor models cannot bethe whole story, and managers need to worry not just about the risks of an overallchange in interest rates but also about shifts in the term structure

3.5 4 4.5 5 5.5 6 6.5 7 7.5

Short-term and long-term interest rates do

not always move in parallel Between

September 1992 and April 2000 short-term

rates rose sharply while long-term rates

declined.

24.4 EXPLAINING THE TERM STRUCTURE

The term structure that we showed in Figure 24.3 was upward-sloping In otherwords, long rates of interest are higher than short rates This is the more commonpattern but sometimes it is the other way around, with short rates higher than longrates Why do we get these shifts in term structure?

Let us look at a simple example Figure 24.3 showed that in the summer of 2001the one-year spot rate was about 3.5 percent The two-year spot rate washigher at 4 percent Suppose that in 2001 you invest in a one-year U.S Treasurystrip You would earn the one-year spot rate of interest and by the end of the yeareach dollar that you invested would have grown to If insteadyou were prepared to invest for two years, you would earn the two-year spot rate

of and by the end of the two years each dollar would have grown to

By keeping your money invested for a further year,your savings grow from $1.0350 to $1.0816, an increase of 4.5 percent This extra 4.5percent that you earn by keeping your money invested for two years rather than

one is termed the forward interest rate or

Notice how we calculated the forward rate When you invest for one year, eachdollar grows to When you invest for two years, each dollar grows to

Therefore, the extra return that you earn for that second year is

In other words, you can think of the two-year investment as earning the one-year spot

rate for the first year and the extra return, or forward rate, for the second year

The Expectations Theory

Would you have been happy in the summer of 2001 to earn an extra 4.5 percent

for investing for two years rather than one? The answer depends on how you

ex-pected interest rates to change over the coming year Suppose, for example, that

you were confident that interest rates would rise sharply, so that at the end of the

year the one-year rate would be 5 percent In that case rather than investing in a

two-year bond and earning the extra 4.5 percent for the second year, you would

do better to invest in a one-year bond and, when that matured, to reinvest the

money for a further year at 5 percent If other investors shared your view, no one

would be prepared to hold the two-year bond and its price would fall It would

stop falling only when the extra return from holding the two-year bond equalled

the expected future one-year rate Let us call this expected rate —that is, the

spot rate of interest at year 1 on a loan maturing at the end of year 2.17Figure 24.7

shows that at that point investors would earn the same expected return from

in-vesting in a two-year loan as from inin-vesting in two successive one-year loans

This is known as the expectations theory of term structure.18It states that in

equi-librium the forward interest rate, , must equal the expected one-year spot rate,

The expectations theory implies that the only reason for an upward-sloping term

structure, such as existed in the summer of 2001, is that investors expect short-term

interest rates to rise; the only reason for a declining term structure is that investors

ex-pect short-term rates to fall.19The expectations theory also implies that investing in

a succession of short-term bonds gives exactly the same expected return as investing

in long-term bonds

If short-term interest rates are significantly lower than long-term rates, it is

of-ten tempting to borrow short-term rather than long-term The expectations theory

1r2

1r2

17

Be careful to distinguish from , the spot interest rate on a two-year bond held from time 0 to time

2 The quantity is a one-year spot rate established at time 1.

18

The expectations theory is usually attributed to Lutz and Lutz See F A Lutz and V C Lutz, The

The-ory of Investment in the Firm, Princeton University Press, Princeton, NJ, 1951.

19

This follows from our example If the two-year spot rate, , exceeds the one-year rate, , then the

for-ward rate, , also exceeds If the forfor-ward rate equals the expected spot rate, then must also

ex-ceed The converse is likewise true.r

An investor can invest either in a

two-year loan [a] or in two successive one-year loans [b] The expectations

theory says that in equilibrium the expected payoffs from these two strategies must be equal In other words, the forward rate, , must equal the expected spot rate, 1r2

f 2

implies that such nạve strategies won’t work If short rates are lower than longrates, then investors must be expecting interest rates to rise When the term struc-ture is upward-sloping, you are likely to make money by borrowing short only if

investors are overestimating future increases in interest rates.

Even on a casual glance the expectations theory does not seem to be the plete explanation of term structure For example, if we look back over the period1926–2000, we find that the return on long-term U.S Treasury bonds was on aver-age 1.9 percent higher than the return on short-term Treasury bills Perhaps short-term interest rates did not go up as much as investors expected, but it seems morelikely that investors wanted a higher expected return for holding long bonds andthat on the average they got it If so, the expectations theory is wrong

com-The expectations theory has few strict adherents, but most economists believethat expectations about future interest rates have an important effect on term struc-ture For example, the expectations theory implies that if the forward rate of inter-est is 1 percent above the spot rate of interest, then your best estimate is that thespot rate of interest will rise by 1 percent In a study of the U.S Treasury bill mar-

ket between 1959 and 1982, Eugene Fama found that a forward premium does on

average precede a rise in the spot rate but the rise is less than the expectations ory would predict.20

the-The Liquidity-Preference the-Theory

What does the expectations theory leave out? The most obvious answer is “risk.”

If you are confident about the future level of interest rates, you will simply choosethe strategy that offers the highest return But, if you are not sure of your forecast,you may well opt for the less risky strategy even if it offers a lower expected return.Remember that the prices of long-duration bonds are more volatile than those

of short-term bonds For some investors this extra volatility may not be a concern.For example, pension funds and life insurance companies with long-term liabili-ties may prefer to lock in future returns by investing in long-term bonds However,

the volatility of long-term bonds does create extra risk for investors who do not

have such long-term fixed obligations

Here we have the basis for the liquidity-preference theory of the term

struc-ture.21If investors incur extra risk from holding long-term bonds, they will mand the compensation of a higher expected return In this case the forward ratemust be higher than the expected spot rate This difference between the forward

de-rate and the expected spot de-rate is usually called the liquidity premium If the

liquidity-preference theory is right, the term structure should be upward-slopingmore often than not Of course, if future spot rates are expected to fall, the term

structure could be downward-sloping and still reward investors for lending long.

But the liquidity-preference theory would predict a less dramatic downwardslope than the expectations theory

20See E F Fama, “The Information in the Term Structure,” Journal of Financial Economics 13 (December

1984), pp 509–528 Evidence from the Treasury bond market that the forward premium has some power

to predict changes in spot rates is provided in J Y Campbell, A W Lo, and A C MacKinlay, The metrics of Financial Markets, Princeton University Press, Princeton, NJ, 1997, pp 421–422.

Econo-21The liquidity-preference hypothesis is usually attributed to Hicks See J R Hicks, Value and Capital:

An Inquiry into Some Fundamental Principles of Economic Theory, 2nd ed., Oxford University Press, ford, 1946 For a theoretical development, see R Roll, The Behavior of Interest Rates: An Application of the Efficient-Market Model to U.S Treasury Bills, Basic Books, Inc., New York, 1970.

Ox-Introducing Inflation

The money cash flows on a U.S Treasury bond are certain, but the real cash flows

are not In other words, Treasury bonds are still subject to inflation risk Let us look

therefore at how uncertainty about inflation affects the risk of bonds with different

maturities.22

Suppose that Irving Fisher is right and short rates of interest always incorporate

fully the market’s latest views about inflation Suppose also that the market learns

more as time passes about the likely inflation rate in a particular year Perhaps

to-day investors have only a very hazy idea about inflation in year 2, but in a year’s

time they expect to be able to make a much better prediction Since investors

ex-pect to learn a good deal about the inflation rate in year 2 from experience in year

1, next year they will be in a much better position to judge the appropriate interest

rate in year 2

You are saving for your retirement Which of the following strategies is the more

risky? Invest in a succession of one-year Treasury bonds or invest in a 20-year bond?

If you buy the 20-year bond, you know what money you will have at the end of

20 years, but you will be making a long-term bet on inflation Inflation may seem

benign now, but who knows what it will be in 10 or 20 years? This uncertainty

about inflation makes it more risky for you to fix today the rates at which you will

lend in the distant future

You can reduce this uncertainty by investing in successive short-term bonds

You do not know the interest rate at which you will be able to reinvest your money

at the end of each year, but at least you know that it will incorporate the latest

in-formation about inflation in the coming year So, if the prospects for inflation

de-teriorate, it is likely that you will be able to reinvest your money at a higher

inter-est rate

Inflation uncertainty may help to explain why long-term bonds provide a

liquid-ity premium If inflation creates additional risks for long-term lenders, borrowers

must offer some incentive if they want investors to lend long Therefore, the forward

rate of interest must be greater than the expected spot rate by an amount that

compensates investors for the extra risk of inflation

Relationships between Bond Returns

These term structure theories tell us how bond prices may be determined at a point

in time More recently, financial economists have proposed some important

theo-ries of how price movements are related These theotheo-ries take advantage of the fact

that the returns on bonds with different maturities tend to move together For

ex-ample, if short-term interest rates are high, it is a good bet that long-term rates will

also be high If short-term rates fall, long-term rates usually keep them company

Such linkages between interest rate movements can tell us something about

rela-tionships between bond prices

The models that bond traders use to exploit these relationships can be quite

complex and we can’t get deeply into the subject here However, the following

ex-ample will give you a flavor of how the models work

Suppose that you can invest in three possible government loans: a

three-month Treasury bill, a medium-term bond, and a long-term bond The return on

E11r22

22See R A Brealey and S M Schaefer, “Term Structure and Uncertain Inflation,” Journal of Finance 32

(May 1977), pp 277–290.

the Treasury bill over the next three months is certain; we will assume it yields a

2 percent quarterly rate The return on each of the other bonds depends on whathappens to interest rates Suppose that you foresee only two possible outcomes—

a sharp rise in interest rates or a sharp fall Table 24.3 summarizes how the prices

of the three investments would be affected Notice that the long-term bond has alonger duration and therefore a wider range of possible outcomes

Here’s the puzzle You know the price of the Treasury bill and the long-termbond But can you get rid of the two question marks in Table 24.3 and figure outwhat the medium-term bond should sell for?

Suppose that you start with $100 You invest half of this money in the Treasurybill and half in the long-term bond In this case the change in the value of your

if interest rates fall Thus, regardless of whether terest rates rise or fall, your portfolio will provide exactly the same payoffs as aninvestment in the medium-term bond Since the two investments provide identi-cal payoffs, they must sell for the same price or there will be a money machine

in-So, the value of the medium-term bond must be halfway between the value of a

Knowing this, you can calculate what the yield to maturity on the medium-termbond has to be You can also calculate its value next year, either

or Everything now checks; regardless of whether interest rates rise or fall, themedium-term bond will provide the same payoff as the package of Treasury billand long-term bond and therefore it must cost the same:

198 ⫹ 1052/2 ⫽ 101.51.5 ⫻ 22 ⫹ 1.5 ⫻ 182 ⫽ ⫹$101.5 ⫻ 22 ⫹ 3.5 ⫻ 1⫺152 4 ⫽ ⫺$6.5

Change in Value Beginning If Interest If Interest Ending

T A B L E 2 4 3

Illustrative payoffs from three

government securities Note the

wider range of outcomes from the

longer-duration loans We don’t

know what the medium-term bond

sells for; we need to figure it out

from how its value changes when

interest rates rise or fall.

Ending Value

Equal holdings (.5 ⫻ 98) ⫹ (.5 ⫻ (.5 ⫻ 100) ⫹ (.5 ⫻ (.5 ⫻ 100) ⫹ (.5 ⫻

& long-term bond

Our example is grossly oversimplified, but you have probably already noticedthat the basic idea is the same that we used when valuing an option To value anoption on a share, we constructed a portfolio of a risk-free loan and the commonstock that would exactly replicate the payoffs from the option That allowed us to